Zero/one-dimensionality of equivariantly graded symplectic cohomology in the Gross–Siebert setup

Determine whether, in the Gross–Siebert toric degeneration framework, the relevant equivariantly graded pieces of symplectic cohomology of the mirror Weinstein domain are either zero or one dimensional, analogous to Proposition 6.6 in GHHPS, to support a proof of mirror symmetry for the Gross–Siebert general fiber.

Background

The proof strategy in GHHPS for mirror symmetry of compact Calabi–Yau hypersurfaces in toric varieties relies on a structural property of symplectic cohomology: certain equivariantly graded pieces are either 0- or 1-dimensional (Prop. 6.6 in GHHPS).

The author notes uncertainty about whether an analogous result holds in the Gross–Siebert setting, which is crucial for extending their method to the general fiber of Gross–Siebert toric degenerations.